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Romashka-Z-Leto [24]
3 years ago
15

Find the locus of a point such that the sum of its distance from the point ( 0 , 2 ) and ( 0 , -2 ) is 6.

Mathematics
2 answers:
jok3333 [9.3K]3 years ago
5 0

Answer:

\displaystyle \frac{x^2}{5}+\frac{y^2}{9}=1

Step-by-step explanation:

We want to find the locus of a point such that the sum of the distance from any point P on the locus to (0, 2) and (0, -2) is 6.

First, we will need the distance formula, given by:

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

Let the point on the locus be P(x, y).

So, the distance from P to (0, 2) will be:

\begin{aligned} d_1&=\sqrt{(x-0)^2+(y-2)^2}\\\\ &=\sqrt{x^2+(y-2)^2}\end{aligned}

And, the distance from P to (0, -2) will be:

\displaystyle \begin{aligned} d_2&=\sqrt{(x-0)^2+(y-(-2))^2}\\\\ &=\sqrt{x^2+(y+2)^2}\end{aligned}

So sum of the two distances must be 6. Therefore:

d_1+d_2=6

Now, by substitution:

(\sqrt{x^2+(y-2)^2})+(\sqrt{x^2+(y+2)^2})=6

Simplify. We can subtract the second term from the left:

\sqrt{x^2+(y-2)^2}=6-\sqrt{x^2+(y+2)^2}

Square both sides:

(x^2+(y-2)^2)=36-12\sqrt{x^2+(y+2)^2}+(x^2+(y+2)^2)

We can cancel the x² terms and continue squaring:

y^2-4y+4=36-12\sqrt{x^2+(y+2)^2}+y^2+4y+4

We can cancel the y² and 4 from both sides. We can also subtract 4y from both sides. This leaves us with:

-8y=36-12\sqrt{x^2+(y+2)^2}

We can divide both sides by -4:

2y=-9+3\sqrt{x^2+(y+2)^2}

Adding 9 to both sides yields:

2y+9=3\sqrt{x^2+(y+2)^2}

And, we will square both sides one final time.

4y^2+36y+81=9(x^2+(y^2+4y+4))

Distribute:

4y^2+36y+81=9x^2+9y^2+36y+36

The 36y will cancel. So:

4y^2+81=9x^2+9y^2+36

Subtracting 4y² and 36 from both sides yields:

9x^2+5y^2=45

And dividing both sides by 45 produces:

\displaystyle \frac{x^2}{5}+\frac{y^2}{9}=1

Therefore, the equation for the locus of a point such that the sum of its distance to (0, 2) and (0, -2) is 6 is given by a vertical ellipse with a major axis length of 3 and a minor axis length of √5, centered on the origin.

LUCKY_DIMON [66]3 years ago
4 0

ANSWER:

We need to know a distance formula to find the locus.

By distance formula,

  • √[x² + (y - 2)²] + √[x² + (y + 2)²] = 6.
  • √(x² + y² - 4y + 4) + √(x² + y² + 4y + 4) = 6.

Let x² + y² + 4 = a.

  • √(a - 4y) + √(a + 4y) = 6.
  • √(a - 4y) = 6 - √(a + 4y).

Squaring both sides,

  • a - 4y = a + 4y + 36 - 12 × √(a + 4y).
  • 8y + 36 = 12 × √(a + 4y).
  • 2y + 9 = 3 × √(a + 4y).

Squaring again,

  • 4y² + 36y + 81 = 9a + 36y.
  • 9a - 4y² - 81 = 0.

Putting back the original value of a,

  • 9x² + 5y² - 45 = 0.

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Answer: 12.65 cm

Step-by-step explanation:

Use the Pythagoras theorem,

a^2 + b^2= c^2

You are given the hypotenuse , which is c and another leg which is either a or b, does not particularly matter- rearrange to find the missing length

c^2 - b^2 = a^2

14^2 - 6^2 = 160

Square root 160 to find the missing length

12.65

Hope this helped :)

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3 years ago
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Step-by-step explanation:

8 0
3 years ago
Sammy bought 15.3 Liters of milk. She poured the milk equally into 5 bottles. There was 0.35 liters of milk left over. Calculate
Zanzabum

Answer:

The capacity of milk in 1 bottle is 2.99 liters

Step-by-step explanation:

Let us solve the question

∵ Sammy bought 15.3 liters of milk

∵ She poured the milk equally into 5 bottles

∵ There were 0.35 liters of milk leftover

→ At first, subtract the leftover from the quantity she bought to

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∴ The capacity of the 5 bottles = 15.3 - 0.35

∴ The capacity of the 5 bottles = 14.95 liters

→ To find the capacity of each bottle divide the total capacity by 5

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8 0
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The population of bacteria in a Petri Dish is growing at a rate of 0.8t^3 + 3.5 thousand per hour. Find the total increase in ba
aliina [53]

Answer:

p=9900\\ bacterias in the initial two hours

Step-by-step explanation:

the growing rate is given by the ecuation

p(t)=0.8(t)^{3} +3.5 thousand per hour

for t=2 we have

p(2)=0.8(2)^{3} +3.5 = 9.9 thousand

p=9900\\ bacterias

In two hours we have 9900 bacterias

3 0
3 years ago
Help please 25 pts. I will mark. One month Justin rented 2 movies and 5 video games for a total of $32. The next month he rented
umka21 [38]

Answer:

The movie costs $3.50 and the video games cost $5.

Step-by-step explanation:

a = 1 movie

b = 1 video game

2a + 5b = $32

8a + 3b = $43

2a x -4 = -8a

5b x -4 = -20b

32 x -4 = -128

   8a + 3b = $43

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--------------------------------

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17b = 85

/17     /17

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----------------------

2a = 7

/2 = /2

a = 3.5

a = $3.50

b = $5

6 0
3 years ago
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